[Paper Review] A variational analysis for the moving finite element method for gradient flows

Xianmin Xu|arXiv (Cornell University)|Sep 3, 2020

Advanced Numerical Methods in Computational Mathematics29 references4 citations

TL;DR

This paper presents a novel variational derivation of the moving finite element method (MFEM) for gradient flow equations using the Onsager principle, ensuring discrete energy dissipation. It proves that MFEM converges to a local minimizer of the energy functional, with optimal convergence rates when the global minimizer is captured, verified numerically for linear and nonlinear diffusion problems including the Allen-Cahn equation.

ABSTRACT

By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.

Motivation & Objective

• To develop a new variational derivation of the moving finite element method (MFEM) for gradient flow systems using the Onsager principle as a foundational framework.
• To establish that the discrete MFEM problem preserves the same energy dissipation structure as the continuous gradient flow system.
• To perform rigorous error analysis for the stationary solution of a nonlinear reaction-diffusion equation using free-knot piecewise polynomial approximation.
• To prove optimal convergence rates under conditions where the discrete scheme detects the global minimizer of the energy functional.
• To validate the theoretical findings with numerical experiments on linear and nonlinear diffusion equations, including the Allen-Cahn equation.

Proposed method

• The Onsager variational principle is applied to derive the time evolution of the system by minimizing a Rayleighian functional composed of dissipation and energy change terms.
• The finite element approximation is formulated in a nonlinear space of free-knot piecewise polynomials, treating both nodal values and mesh vertices as unknowns to be optimized.
• A system of ordinary differential equations (ODEs) is derived for the time evolution of nodal values and mesh points, which matches the classical MFEM formulation without requiring mollification or singular function handling.
• The energy dissipation structure of the continuous problem is preserved in the discrete formulation, enabling stability and convergence analysis.
• Nonlinear approximation theory for free-knot piecewise polynomials is used to derive error bounds, particularly for the stationary solution.
• Numerical experiments solve the discrete system until energy decrease per step falls below a tolerance, with adaptive mesh refinement observed to concentrate nodes in regions of high solution curvature.

Experimental results

Research questions

• RQ1 Can the Onsager variational principle be used as a systematic framework to derive the moving finite element method for gradient flow systems?
• RQ2 Does the discrete MFEM formulation preserve the energy dissipation structure of the continuous gradient flow?
• RQ3 Under what conditions does the MFEM converge to a local minimizer of the energy functional as time tends to infinity?
• RQ4 What is the optimal convergence rate of the MFEM for the stationary solution when the global minimizer is detected in the discrete space?
• RQ5 How do the error in the H¹ norm and the energy error scale with mesh refinement in the MFEM for nonlinear gradient flows?

Key findings

• The MFEM derived via the Onsager principle exactly preserves the energy dissipation structure of the continuous gradient flow system.
• Under suitable conditions, the MFEM solution converges to a local minimizer of the energy functional as time approaches infinity.
• When the global minimizer is detected in the discrete free-knot piecewise linear space, the H¹ error converges at the optimal rate of O(N⁻¹) with respect to the number of elements N.
• The energy error converges at the optimal rate of O(N⁻²), which is consistent with the theoretical bounds derived from nonlinear approximation theory.
• Numerical results for the Allen-Cahn equation with ε = 0.05 and ε = 0.01 confirm optimal convergence rates: H¹ error ≈ O(N⁻¹) and energy error ≈ O(N⁻²), even on coarse meshes with N = 5.
• The adaptive mesh naturally concentrates nodes in the interior layer of sharp transition profiles, indicating uniform error distribution in the H¹ norm.